Irish Math. Soc. Bulletin
Number 72, Winter 2013, 33–34
ISSN 0791-5578
UNIVERSALITY AND POTENTIAL THEORY
MYRTO MANOLAKI
This is an abstract of the PhD thesis Universality and potential
theory written by Myrto Manolaki under the supervision of Prof.
Stephen J. Gardiner at the School of Mathematical Sciences, University College Dublin, and submitted in May 2013.
Universality is an abstract notion which relates to various mathematical contexts. Generally speaking, an object is called universal if, via a countable process, it can approximate every object in
some class. This thesis investigates one of the most widely-studied
instances of universality: the case of universal series, where the approximation is obtained by considering subsequences of partial sums.
In particular, it focuses on the study of harmonic functions on domains in RN , and holomorphic functions on planar domains, which
possess universal polynomial expansions.
The first part of the thesis is concerned with the related phenomenon of overconvergence, which is of independent interest. Ostrowski
showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside
the disc of convergence. In Chapter 3 we investigate the corresponding problem for the homogeneous polynomial expansion of a
harmonic function and we obtain analogues of the three main theorems of Ostrowski. We further show that the results for harmonic
functions display new features in the case of higher dimensions.
Let Ω be a domain in RN and let w ∈ Ω. Chapter 4 is concerned
with the class U (Ω, w) of harmonic functions on Ω which are universal about w, in the sense that subsequences of partial sums of
the homogeneous polynomial expansion approximate arbitrary harmonic polynomials on arbitrary compact sets in RN \ Ω that have
2010 Mathematics Subject Classification. 31B05, 30K05.
Key words and phrases. harmonic function; harmonic approximation; homogeneous harmonic polynomials; overconvergence; universal series.
Received on 14-12-2013.
Support from Science Foundation Ireland through the Research Frontiers Programme (project number 09/RFP/MTH2149) is gratefully acknowledged.
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M. MANOLAKI
connected complement. It is known that, if RN ∪ {∞} \ Ω is connected, then for each w ∈ Ω, the class U (Ω, w) is non-empty. Under
the assumption that Ω omits an infinite
cone, we show that the conN
nectedness hypothesis on R ∪ {∞} \ Ω is essential, and that a
harmonic function which is universal at one point is actually universal at all points of Ω. Further, we conclude that no function in
U (Ω, w) has a harmonic extension to a larger domain.
The other main component of the thesis is concerned with the class
of holomorphic functions on planar domains which possess universal
Taylor series. We adopt a potential theoretical approach in order
to shed some light on the main questions of the area. In particular,
we use various potential theoretical notions such as harmonic measure, the Martin boundary and minimal thinness, in order to address
questions concerning the existence (Chapter 5) and the boundary
behaviour (Chapter 6) of such functions. Our main result is that
holomorphic functions possessing universal Taylor series are always
unbounded.
Finally, some further work is presented in Chapters 7 and 8, which
involve some results on potential theory and universal Laurent series.
The main results of this thesis are included in [1, 2, 3].
References
[1] S. J. Gardiner and M. Manolaki, Boundary behaviour of universal Taylor series
on multiply connected domains, submitted.
[2] M. Manolaki, Ostrowski-type theorems for harmonic functions, J. Math. Anal.
Appl. 391 (2012), 480-488.
[3] M. Manolaki, Universal polynomial expansions of harmonic functions, Potential Anal. 38 (2013), 985-1000.
Department of Mathematics, University of Western Ontario, London, ON N6A 3K7, Canada
E-mail address, M. Manolaki: arhimidis8@yahoo.gr